multicut is the generalization of mincut where,
instead of \(s\)  and \(t\) , the input contains a set \((s_1,t_1)\) , \(\dots \) ,
\((s_\ell ,t_\ell )\)  of terminal pairs. The task is to find a set \(S\) 
of at most \(k\)  nonterminal vertices that separate \(s_i\)  and \(t_i\)  for
every \(1\le i \le \ell \) . multicut is known to be fpt
[1]}, [2]} parameterized by \(k\)  and \(\ell \) .
In the \(\mathcal {G}\) -multicut problem, we additionally require that
\(S\)  induces a graph from \(\mathcal {G}\) . It is not difficult to generalize
Theorem  for \(\mathcal {G}\) -multicut: all we need to do
is to change the construction of \(\varphi \)  such that it requires the
separation of each pair \((s_i,t_i)\) .
We state this here in an even more general form.
In the \(\mathcal {G}\) -multicut-uncut problem the input contains an
additional integer \(\ell ^{\prime } \le \ell \) , and we change the problem by requiring for
every \(\ell ^{\prime } \le i \le \ell \)  that \(S\)  does not separate \(s_i\) 
and \(t_i\) .
